cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 26 results. Next

A045711 Primes with first digit 5.

Original entry on oeis.org

5, 53, 59, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 5003, 5009, 5011, 5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087, 5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179, 5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261
Offset: 1

Views

Author

Felice Russo

Keywords

Comments

Subsequence of A000040.

Links

Crossrefs

Column k=5 of A262369.
For primes with initial digit d (1 <= d <= 9) see A045707, A045708, A045709, A045710, A045711, A045712, A045713, A045714, A045715; A073517, A073516, A073515, A073514, A073513, A073512, A073511, A073510, A073509.

Programs

  • Magma
    [p: p in PrimesUpTo(5300) | Intseq(p)[#Intseq(p)] eq 5]; // Vincenzo Librandi, Aug 08 2014
  • Mathematica
    Select[Table[Prime[n], {n, 5300}], First[IntegerDigits[#]]==5 &] (* Vincenzo Librandi, Aug 08 2014 *)
  • Python
    from itertools import chain, count, islice
from sympy import primerange
def A045711_gen(): # generator of terms
    return chain.from_iterable(primerange(5*(m:=10**l),6*m) for l in count(0))
A045711_list = list(islice(A045711_gen(),40)) # Chai Wah Wu, Dec 08 2024
  • Python
    from sympy import primepi
def A045711(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x+primepi(min(5*(m:=10**(l:=len(str(x))-1))-1,x))-primepi(min(6*m-1,x))+sum(primepi(5*(m:=10**i)-1)-primepi(6*m-1) for i in range(l))
    return bisection(f,n,n) # Chai Wah Wu, Dec 08 2024

Extensions

More terms from Erich Friedman.
Leading 5 added by Jaroslav Krizek, Apr 29 2010

A131835 Numbers starting with 1.

Original entry on oeis.org

1, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142
Offset: 1

Views

Author

Andrew Good (yipes_stripes(AT)yahoo.com), Jul 20 2007

Keywords

Comments

The lower and upper asymptotic densities of this sequence are 1/9 and 5/9, respectively. - Amiram Eldar, Feb 27 2021

Links

Crossrefs

Subsequence of A011531.
Disjoint union of A045707 and A206286.
Cf. A000030, A000027, A002275, A262390 (permutation).
Cf. A217394, A217395, A217397, A217398, A217399, A217400, A217401, A217402.

Programs

  • Haskell
    a131835 n = a131835_list !! (n-1)
a131835_list = concat $
               iterate (concatMap (\x -> map (+ 10 * x) [0..9])) [1]
-- Reinhard Zumkeller, Jul 16 2014
  • Maple
    isA131835 := proc(n) if op(-1,convert(n,base,10)) = 1 then true; else false ; fi ; end: for n from 1 to 300 do if isA131835(n) then printf("%d, ",n) ; fi ; od : # R. J. Mathar, Jul 24 2007
  • Mathematica
    Select[Range[150], IntegerDigits[#][[1]] == 1 &] (* Amiram Eldar, Feb 27 2021 *)
  • PARI
    a(n, {base=10}) = my (o=1); while (n>o, n-=o; o*=base); return (o+n-1) \\ Rémy Sigrist, Jun 23 2017
  • PARI
    a(n) = n--; s = #digits(9*n+1); n + 8 * (10^(s-1))/9 + 1/9 \\ David A. Corneth, Jun 23 2017
  • PARI
    nxt(n) = my(d = digits(n+1)); if(d[1]==1, n+1, 10^#d) \\ David A. Corneth, Jun 23 2017
  • Python
    def A131835(n): return n+(10**(len(str(9*n-8))-1)<<3)//9 # Chai Wah Wu, Dec 07 2024

Formula

A000030(a(n)) = 1. - Reinhard Zumkeller, Jul 16 2014
a(A002275(n)+1) = 10^n for any n >= 0. - Rémy Sigrist, Jun 23 2017
a(n) = n + (8*10^floor(log_10(9*n-8))-8)/9. - Alan Michael Gómez Calderón, May 16 2023

Extensions

More terms from R. J. Mathar, Jul 24 2007

A045708 Primes with first digit 2.

Original entry on oeis.org

2, 23, 29, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221
Offset: 1

Views

Author

Felice Russo

Keywords

Links

Crossrefs

Cf. A000040.
For primes with initial digit d (1 <= d <= 9) see A045707, A045708, A045709, A045710, A045711, A045712, A045713, A045714, A045715; A073517, A073516, A073515, A073514, A073513, A073512, A073511, A073510, A073509
Cf. A000030, subsequence of A208272.
Column k=2 of A262369.

Programs

  • Haskell
    a045708 n = a045708_list !! (n-1)
a045708_list = filter ((== 2) . a000030) a000040_list
-- Reinhard Zumkeller, Mar 16 2012
  • Magma
    [p: p in PrimesUpTo(2300) | Intseq(p)[#Intseq(p)] eq 2]; // Vincenzo Librandi, Aug 08 2014
  • Mathematica
    Select[Table[Prime[n], {n, 3000}], First[IntegerDigits[#]]==2 &] (* Vincenzo Librandi, Aug 08 2014 *)
  • Python
    from sympy import isprime
def agen(limit=float('inf')):
  yield 2
  digits, adder = 1, 20
  while True:
    for i in range(1, 10**digits, 2):
      test = adder + i
      if test > limit: return
      if isprime(test): yield test
    digits, adder = digits+1, adder*10
agento = lambda lim: agen(limit=lim)
print(list(agento(2222))) # Michael S. Branicky, Feb 23 2021
  • Python
    from sympy import primepi
def A045708(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x+primepi(min(((m:=10**(l:=len(str(x))-1))<<1)-1,x))-primepi(min(3*m-1,x))+sum(primepi(((m:=10**i)<<1)-1)-primepi(3*m-1) for i in range(l))
    return bisection(f,n,n) # Chai Wah Wu, Dec 07 2024

Formula

See A045707 for comments on density of these sequences.

Extensions

More terms from Erich Friedman.
Offset fixed by Reinhard Zumkeller, Mar 15 2012

A045709 Primes with first digit 3.

Original entry on oeis.org

3, 31, 37, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229
Offset: 1

Views

Author

Felice Russo

Keywords

Links

Crossrefs

Cf. A000040.
For primes with initial digit d (1 <= d <= 9) see A045707, A045708, A045709, A045710, A045711, A045712, A045713, A045714, A045715; A073517, A073516, A073515, A073514, A073513, A073512, A073511, A073510, A073509.
Column k=3 of A262369.

Programs

  • Magma
    [p: p in PrimesUpTo(3300) | Intseq(p)[#Intseq(p)] eq 3]; // Vincenzo Librandi, Aug 08 2014
  • Mathematica
    Select[Table[Prime[n], {n, 4000}], First[IntegerDigits[#]]==3 &] (* Vincenzo Librandi, Aug 08 2014 *)
  • PARI
    isok(n) = isprime(n) && (digits(n, 10)[1] == 3) \\ Michel Marcus, Jun 08 2013
  • Python
    from itertools import chain, count, islice
from sympy import primerange
def A045709_gen(): # generator of terms
    return chain.from_iterable(primerange(3*(m:=10**l),m<<2) for l in count(0))
A045709_list = list(islice(A045709_gen(),40)) # Chai Wah Wu, Dec 07 2024
  • Python
    from sympy import primepi
def A045709(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x+primepi(min(3*(m:=10**(l:=len(str(x))-1))-1,x))-primepi(min((m<<2)-1,x))+sum(primepi(3*(m:=10**i)-1)-primepi((m<<2)-1) for i in range(l))
    return bisection(f,n,n) # Chai Wah Wu, Dec 07 2024

Extensions

More terms from Erich Friedman.

A045714 Primes with first digit 8.

Original entry on oeis.org

83, 89, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 8009, 8011, 8017, 8039, 8053, 8059, 8069, 8081, 8087, 8089, 8093, 8101, 8111, 8117, 8123, 8147, 8161, 8167, 8171, 8179, 8191, 8209, 8219, 8221, 8231, 8233, 8237, 8243
Offset: 1

Views

Author

Felice Russo

Keywords

Links

Crossrefs

For primes with initial digit d (1 <= d <= 9) see A045707, A045708, A045709, A045710, A045711, A045712, A045713, A045714, A045715; A073517, A073516, A073515, A073514, A073513, A073512, A073511, A073510, A073509.
Column k=8 of A262369.

Programs

  • Magma
    [p: p in PrimesUpTo(10^4) | Intseq(p)[#Intseq(p)] eq 8]; // Bruno Berselli, Jul 19 2014
  • Mathematica
    Flatten[Table[Prime[Range[PrimePi[8 * 10^n] + 1, PrimePi[9 * 10^n]]], {n, 3}]] (* Alonso del Arte, Jul 19 2014 *)
  • Python
    from itertools import chain, count, islice
from sympy import primerange
def A045714_gen(): # generator of terms
    return chain.from_iterable(primerange((m:=10**l)<<3,9*m) for l in count(0))
A045714_list = list(islice(A045714_gen(),40)) # Chai Wah Wu, Dec 08 2024
  • Python
    from sympy import primepi
def A045714(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x+primepi(min(((m:=10**(l:=len(str(x))-1))<<3)-1,x))-primepi(min(9*m-1,x))+sum(primepi(((m:=10**i)<<3)-1)-primepi(9*m-1) for i in range(l))
    return bisection(f,n,n) # Chai Wah Wu, Dec 08 2024

Extensions

More terms from Erich Friedman.

A073517 Number of primes less than 10^n with initial digit 1.

Original entry on oeis.org

0, 4, 25, 160, 1193, 9585, 80020, 686048, 6003530, 53378283, 480532488, 4369582734, 40063566855, 369893939287, 3435376839800, 32069022099022, 300694113015105, 2830466318006780, 26735673312004455, 253315661161665338, 2406763761677705769, 22923886160712831134, 218839439542390117580
Offset: 1

Views

Author

Shyam Sunder Gupta, Aug 14 2002

Keywords

Examples

			a(2)=4 because there are 4 primes up to 10^2 whose initial digit is 1 (11, 13, 17 and 19).

Links

Crossrefs

Cf. A000720 (pi), A073509 to A073517, their sum is A006880.
For primes with initial digit d (1 <= d <= 9) see A045707, A045708, A045709, A045710, A045711, A045712, A045713, A045714, A045715; A073517, A073516, A073515, A073514, A073513, A073512, A073511, A073510, A073509.

Programs

  • Mathematica
    f[n_] := f[n] = PrimePi[2*10^n] - PrimePi[10^n] + f[n - 1]; f[0] = 0; Table[ f[n], {n, 0, 13}]
  • PARI
    a(n,d=1)=sum(k=0, n-1, primepi((d+1)*10^k-1) - primepi(d*10^k-1)) \\ Andrew Howroyd, Dec 15 2024

Formula

a(n) = Sum_{k=0..n-1} pi(2*10^k-1) - pi(10^k-1). - Andrew Howroyd, Dec 15 2024

Extensions

Edited and extended by Robert G. Wilson v, Aug 29 2002
a(21)-a(22) added by David Baugh, Mar 21 2015
a(23) from Chai Wah Wu, Sep 18 2018
Offset corrected by Andrew Howroyd, Dec 15 2024

A045710 Primes with first digit 4.

Original entry on oeis.org

41, 43, 47, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057, 4073, 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, 4153, 4157, 4159, 4177, 4201, 4211
Offset: 1

Views

Author

Felice Russo

Keywords

Links

Crossrefs

For primes with initial digit d (1 <= d <= 9) see A045707, A045708, A045709, A045710, A045711, A045712, A045713, A045714, A045715; A073517, A073516, A073515, A073514, A073513, A073512, A073511, A073510, A073509.
Column k=4 of A262369.

Programs

  • Magma
    [p: p in PrimesUpTo(10^4) | Intseq(p)[#Intseq(p)] eq 4]; // Bruno Berselli, Jul 19 2014
  • Mathematica
    Select[Table[Prime[n], {n, 1000}], First[IntegerDigits[#]] == 4 &]
Flatten[Table[Prime[Range[PrimePi[4 * 10^n] + 1, PrimePi[5 * 10^n]]], {n, 3}]] (* Alonso del Arte, Jul 19 2014 *)
  • Python
    from itertools import chain, count, islice
from sympy import primerange
def A045710_gen(): # generator of terms
    return chain.from_iterable(primerange((m:=10**l)<<2,5*m) for l in count(0))
A045710_list = list(islice(A045710_gen(),40)) # Chai Wah Wu, Dec 08 2024
  • Python
    from sympy import primepi
def A045710(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x+primepi(min(((m:=10**(l:=len(str(x))-1))<<2)-1,x))-primepi(min(5*m-1,x))+sum(primepi(((m:=10**i)<<2)-1)-primepi(5*m-1) for i in range(l))
    return bisection(f,n,n) # Chai Wah Wu, Dec 08 2024

Extensions

More terms from Erich Friedman.

A045712 Primes with first digit 6.

Original entry on oeis.org

61, 67, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 6007, 6011, 6029, 6037, 6043, 6047, 6053, 6067, 6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133, 6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, 6229
Offset: 1

Views

Author

Felice Russo

Keywords

Links

Crossrefs

For primes with initial digit d (1 <= d <= 9) see A045707, A045708, A045709, A045710, A045711, A045712, A045713, A045714, A045715; A073517, A073516, A073515, A073514, A073513, A073512, A073511, A073510, A073509.
Column k=6 of A262369.

Programs

  • Magma
    [p: p in PrimesUpTo(10^4) | Intseq(p)[#Intseq(p)] eq 6]; // Bruno Berselli, Jul 19 2014
  • Mathematica
    Flatten[Table[Prime[Range[PrimePi[6 * 10^n] + 1, PrimePi[7 * 10^n]]], {n, 3}]] (* Alonso del Arte, Jul 19 2014 *)
Select[Table[Prime[n],{n, 7000}], First[IntegerDigits[#]]==6 &] (* Vincenzo Librandi, Aug 08 2014 *)
  • Python
    from itertools import chain, count, islice
from sympy import primerange
def A045712_gen(): # generator of terms
    return chain.from_iterable(primerange(6*(m:=10**l),7*m) for l in count(0))
list(islice(A045712_gen(),40)) # Chai Wah Wu, Dec 08 2024
  • Python
    from sympy import primepi
def A045712(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x+primepi(min(6*(m:=10**(l:=len(str(x))-1))-1,x))-primepi(min(7*m-1,x))+sum(primepi(6*(m:=10**i)-1)-primepi(7*m-1) for i in range(l))
    return bisection(f,n,n) # Chai Wah Wu, Dec 08 2024

Extensions

More terms from Erich Friedman.

A073509 Number of primes less than 10^n with initial digit 9.

Original entry on oeis.org

0, 1, 15, 127, 1006, 8230, 70320, 614821, 5453140, 48982456, 444608278, 4070532710, 37535715441, 348245215460, 3247889171908, 30429496751905, 286235215995588, 2702000272361599, 25586688305447928, 242978340446949438, 2313264023790027111, 22074118786158858975
Offset: 1

Views

Author

Shyam Sunder Gupta, Aug 14 2002

Keywords

Examples

			a(2) = 1 because there is 1 prime less than 100 whose initial digit is 9, i.e., 97.

Links

Crossrefs

A006880(n) = A073509(n)+ ... + A073516(n)+A073517(n-1).
For primes with initial digit d (1 <= d <= 9) see A045707, A045708, A045709, A045710, A045711, A045712, A045713, A045714, A045715; A073517, A073516, A073515, A073514, A073513, A073512, A073511, A073510, A073509

Programs

  • Mathematica
    f[n_] := f[n] = PrimePi[10^(n + 1)] - PrimePi[9*10^n] + f[n - 1]; f[0] = 0; Table[f[n], {n, 0, 12}]

Extensions

Edited and extended by Robert G. Wilson v, Aug 29 2002
a(20)-a(22) added by David Baugh, Mar 22 2015

A073510 Number of primes less than 10^n with initial digit 8.

Original entry on oeis.org

0, 2, 17, 127, 1003, 8326, 71038, 618610, 5481646, 49221187, 446590932, 4087194991, 37677478288, 349465615584, 3258501713644, 30522628848972, 287059041039078, 2709339704446862, 25652489700275636, 243571629996128384, 2318640708958531064, 22123070798400775157
Offset: 1

Views

Author

Shyam Sunder Gupta, Aug 14 2002

Keywords

Examples

			a(2)=2 because there are 2 primes up to 10^2 whose initial digit is 8 (namely 83 and 89).

Links

Crossrefs

A006880(n) = A073509(n)+ ... + A073516(n)+A073517(n-1).
For primes with initial digit d (1 <= d <= 9) see A045707, A045708, A045709, A045710, A045711, A045712, A045713, A045714, A045715; A073517, A073516, A073515, A073514, A073513, A073512, A073511, A073510, A073509

Programs

  • Mathematica
    f[n_] := f[n] = PrimePi[9*10^n] - PrimePi[8*10^n] + f[n - 1]; f[0] = 0; Table[ f[n], {n, 0, 12}]

Extensions

Edited and extended by Robert G. Wilson v, Aug 29 2002
a(20)-a(22) added by David Baugh, Mar 22 2015
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